For every closed Riemannian manifold , there exists a positive number such that for any there exists some such that for every metric space with Gromov–Hausdorff distance to less than , the Vietoris–Rips complex is homotopy equivalent to .
Key points:
The result is qualitative
the threshold depends solely on the geometry of . But the theorem did not say how!
is a function (probably a fraction) of .
Nonetheless, the result answers Hausmann’s question!
Hausmann, Jean-Claude. 1995. “On the Vietoris-RipsComplexes and a CohomologyTheory for MetricSpaces.” In Prospects in Topology (AM-138), 175–88. Princeton University Press.
Komendarczyk, Rafal, Sushovan Majhi, and Will Tran. 2024. “Topological Stability and Latschev-Type Reconstruction Theorems for Spaces.”
Latschev, J. 2001. “Vietoris-Rips Complexes of Metric Spaces Near a Closed Riemannian Manifold.”Archiv Der Mathematik 77 (6): 522–28. https://doi.org/10.1007/PL00000526.
———. 2023b. “VietorisRips Complexes of Metric Spaces Near a Metric Graph.”Journal of Applied and Computational Topology, May. https://doi.org/10.1007/s41468-023-00122-z.